If the limits of nets in a topological space $X$ are unique, then $X$ is hausdorff.

Question

I looked at a proof of the statement in the title, and I got a question about it...

In the proof, it begins with assuming that $X$ is not hausdorff.

Then, there exists $x,y\in X$, w/ $x\neq y$, s.t. $\forall U,V\in\mathcal{T}_X$, w/ $x\in U$ and $y\in V$, and $U\cap V\neq\emptyset$.

And then it considers $\Lambda=\{(U,V)\mid U~\text{is a nbd of}~x,~V~\text{is a nbd of}~y,~\text{and}~U\cap V\neq\emptyset\}$, and it claims that $\Lambda$ is a directed set with the preorder $\prec$ on $\Lambda$ defined by $(U,V)\prec(W,N)\iff U\supset W,~\text{and}~V\supset N$.

My problem is that I cannot convince myself why $\forall\lambda_1,\lambda_2\in\Lambda$, $\exists\lambda\in\Lambda$ such that $\lambda_1\prec\lambda$ and $\lambda_2\prec\lambda$. In particular, how one should construct such $\lambda$?

Thanks in advance.

Answer 1

Let $\lambda_i=\langle U_i,V_i\rangle\in\Lambda$ for $i=1,2$. Then $\lambda=\langle U_1\cap U_2,V_1\cap V_2\rangle\in\Lambda$, and $\lambda_i\prec\lambda$ for $i=1,2$.